It is always a pleasure to hear from someone finding something new that they really like, particularly when I share the sentiment. I think that whatever direction you go from here, it will take listening to a wide variety of alternatives because there really is no such thing as a "tube" sound; there is a wide variety of sounds generated by tube gear and the "extremes" in tube gear are much wider apart than the extremes in top end solid state. Tube gear will vary from being very rich and somewhat lacking in "detail" or sharp attack of hard transients or it may be just the opposite and can sound far more brittle and edgy than almost any solid state stuff. It is a matter of finding what you like and what fits your set of components.
There is a lot of good advice above about staying with what you have, at least for now, and trying different tubes and certain upgraded parts. My only caveat on the parts is that one should not necessarily expect an "improvement" by dropping in expensive parts--each part has a particular "voice" that may or may not be the direction you want to take.
The necessity for experimentation, and the relative ease in swapping parts in most tube gear may either be a source of fun and enjoyment, or extremely frustrating, it depends on your temperament and commitment to the chase.
There are several broad areas that should be explored before you focus on a particular area (obsession). If you are not totally committed to a particular set of speakers, the exploration should involve hearing high-powered vs. low-powered tube gear. To me, the difference between a typical high-powered amp and a low-powered amp (under 10 watts) is wide and I like the low-powered stuff enough to accept the extreme limitation on the choice of speakers. There is also the matter of looking into single-ended vs. pushpull amps; I like examples of both, but, there are a lot of people that only like single-ended amps. There is another kind of amp--output transformerless (OTL) that is a MUST hear--they are extremely dynamic and immediate sounding in a way that no other type of amp, solid state or tube, can match.
Good luck on the quest.